This question examines the addition of a (synthetic) echo to the voice recording, such as would be found in a reverberant room. To do this, use a difference equation of the form

y(n)   =  x(n) + αy(n - D)

where x(n) is the input audio, y(n) is the output (echoed) audio, D is the echo delay (in samples), and α governs the amount of echo fed back.

Explain how to convert the above equation into a form suitable for passing to MATLAB's filter() command. Use α = 0.8 and D = 2 samples to explain your reasoning.

A delay of 2 samples (as in the previous part) would not be audible. Select an echo delay of 0.2 seconds, and α = 0.4. Using your reasoning above, implement the reverberation equation, and listen to the result.

Experiment with different values of the parameters α and D.  In your report, include a plot of the waveform, and explain in your own words the physical significance of these parameters.

Convert the difference equation to a z transfer function. Where are the poles located?

Suppose the equation governing the reverberation is

y(n)   =  x(n) + αx(n - D)

What would be the physical significance of this form, as opposed to that used in equation 4? Implement an audio echo system based on equation 5, and listen to the results.